definition of neighborhood of subset
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of topological space.
- The reader knows a definition of open set.
Target Context
- The reader will have a definition of neighborhood of subset.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( T\): \(\in \{\text{ the topological spaces }\}\)
\( S\): \(\subseteq T\)
\(*N_S\): \(: \subseteq T\)
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Conditions:
\(\exists U_S \in \{\text{ the open subsets of } T\} (S \subseteq U_S \subseteq N_S)\)
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2: Note
Some people mean 'open neighborhood' by 'neighborhood', while in many cases, whether it is a (not-necessarily-open) neighborhood or an open neighborhood does not matter, because, for example, "there is a neighborhood" equals "there is an open neighborhood", because if there is a neighborhood, there is an open neighborhood in it, and if there is an open neighborhood, there is the neighborhood that is the open neighborhood.
